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Slenderness Effects for Concrete Columns in Sway Frame - Moment Magnification Method
Version: July-19-2017
Slender Concrete Column Design in Sway Frame Buildings
Evaluate slenderness effect for columns in a sway frame multistory reinforced concrete building by designing the first
story exterior column. The clear height of the first story is 13 ft-4 in., and is 10 ft-4in. for all of the other stories.
Lateral load effects on the building are governed by wind forces. Compare the calculated results with the values
presented in the Reference and with exact values from spColumn engineering software program from StructurePoint.
Figure 1 – Reinforced Concrete Column Cross-Section
Version: July-19-2017
Contents
1. Factored Axial Loads and Bending Moments .......................................................................................................... 2
1.1. Service loads ..................................................................................................................................................... 2
1.2. Load Combinations – Factored Loads .............................................................................................................. 2
2. Slenderness Effects and Sway or Nonsway Frame Designation .............................................................................. 3
3. Determine Slenderness Effects ................................................................................................................................. 4
4. Moment Magnification at Ends of Compression Member ....................................................................................... 5
5. Moment Magnification along Length of Compression Member ............................................................................ 11
6. Column Design....................................................................................................................................................... 16
6.1. c, a, and strains in the reinforcement .............................................................................................................. 16
6.2. Forces in the concrete and steel....................................................................................................................... 17
6.3. ϕPn and ϕMn .................................................................................................................................................... 17
7. Column Interaction Diagram - spColumn Software ............................................................................................... 19
8. Summary and Comparison of Design Results ........................................................................................................ 28
9. Conclusions & Observations .................................................................................................................................. 30
1
Code
Building Code Requirements for Structural Concrete (ACI 318-14) and Commentary (ACI 318R-14)
Reference
Notes on ACI 318-11 Building Code Requirements for Structural Concrete, Twelfth Edition, 2013 Portland
Cement Association, Example 11-2
Design Data
fc’ = 6,000 psi for columns in the bottom two stories
= 4,000 psi elsewhere
fy = 60,000 psi
Slab thickness = 7 in.
Exterior Columns = 22 in. x 22 in.
Interior Columns = 24 in. x 24 in.
Beams = 24 in. x 20 in. x 24 ft
Superimposed dead load = 30 psf
Roof live load = 30 psf
Floor live load = 50 psf
Wind loads computed according to ASCE 7-10
Total building loads in the first story from structural analysis:
D = 17,895 kip
L = 1,991 kip
Lr = 270 kip
W = 0 kip, wind loads in the story cause compression in some columns and tension in others and thus would
cancel out.
2
1. Factored Axial Loads and Bending Moments
1.1. Service loads
Table 1 - Exterior column service loads
Load Case Axial Load,
kip
Bending Moment, ft-kip
Top Bottom
Dead, D 622.4 34.8 17.6
Live, L 73.9 15.4 7.7
Roof Live, Lr 8.6 0.0 0.0
Wind, W (N-S) -48.3 17.1 138.0
Wind, W (S-N) 48.3 -17.1 -138.0
1.2. Load Combinations – Factored Loads ASCE 7-10 (2.3.2)
Table 2 - Exterior column factored loads
ASCE 7-10
Reference No. Load Combination
Axial Load,
kip
Bending Moment, ft-kip MTop,ns
ft-kip
MBottom,ns
ft-kip
MTop,s
ft-kip
MBottom,s
ft-kip
Top Bottom
2.3.2-1 1 1.4D 871.4 48.7 24.6 48.7 24.6 --- ---
2.3.2-2 2 1.2D + 1.6L + 0.5Lr 869.4 66.4 33.4 66.4 33.4 --- ---
2.3.2-3
3 1.2D + 0.5L + 1.6 Lr 797.6 49.5 25.0 49.5 25.0 --- ---
4 1.2D + 1.6Lr + 0.8W 722.0 55.4 131.5 41.8 21.1 13.7 110.4
5 1.2D + 1.6Lr - 0.8W 799.3 28.1 -89.3 41.8 21.1 -13.7 -110.4
2.3.2-4 6 1.2D + 0.5L + 0.5Lr + 1.6W 710.9 76.8 245.8 49.5 25.0 27.4 220.8
7 1.2D + 0.5L + 0.5Lr - 1.6W 865.4 22.1 -195.8 49.5 25.0 -27.4 -220.8
2.3.2-6 8 0.9D + 1.6W 482.9 58.7 236.6 31.3 15.8 27.4 220.8
9 0.9D - 1.6W 637.4 4.0 -205.0 31.3 15.8 -27.4 -220.8
3
2. Slenderness Effects and Sway or Nonsway Frame Designation
Columns and stories in structures are considered as non-sway frames if the increase in column end moments due to
second-order effects does not exceed 5% of the first-order end moments, or the stability index for the story (Q) does
not exceed 0.05. ACI 318-14 (6.6.4.3)
∑Pu is the total vertical load in the first story corresponding to the lateral loading case for which ∑Pu is greatest
(without the wind loads, which would cause compression in some columns and tension in others and thus would cancel
out). ACI 318-14 (6.6.4.4.1 and R6.6.4.3)
Vus is the factored horizontal story shear in the first story corresponding to the wind loads, and Δo is the first-order
relative deflection between the top and bottom of the first story due to Vu. ACI 318-14 (6.6.4.4.1 and R6.6.4.3)
From Table 2, load combinations (2.3.2-4 No. 5 and 6) provide the greatest value of ∑Pu.
1.2 0.5 0.5 1.2 17,895 0.5 1,991 0.5 270 22,605 kipu rP D L L ASCE 7-10 (2.3.2-4)
1.6 1.6 302.6 484.2 kipus sV V ASCE 7-10 (2.3.2-6)
1.6 1.6 0.28 0 0.45 in.o
22,605 0.450.12 0.05
484.2 15 12 20 / 2
u o
us c
PQ
V l
ACI 318-14 (Eq. 6.6.4.4.1)
Thus, the frame at the first story level is considered sway.
4
3. Determine Slenderness Effects
4 4422
0.7 0.7 13,665 in.12 12
column
cI ACI 318-14 (Table 6.6.3.1.1(a))
'57,000 57,000 6000 4,415 ksic cE f ACI 318-14 (19.2.2.1.b)
For the column below level 2:
34,415 13,665355 10 in.kip
15 12 20 / 2
c column
c
E I
l
For the column above level 2:
34,415 13,665419 10 in.kip
12 12
c column
c
E I
l
For beams framing into the columns:
33,605 5,60070 10 in.kip
24 12
b beam
b
E I
l
Where:
'57,000 57,000 4000 3,605 ksib cE f ACI 318-14 (19.2.2.1.b)
3 3424 20
0.35 0.35 5,600 in.12 12
beam
b hI
ACI 318-14 (Table 6.6.3.1.1(a))
355 41911
70
c columns
A
beams
EI
l
EI
l
ACI 318-14 (Figure R6.2.5)
1.0B (Column essentially fixed at base) ACI 318-14 (Figure R6.2.5)
Using Figure R6.2.5 from ACI 318-14 k = 1.9 as shown in the figure below for the exterior columns with one
beam framing into them in the directions of analysis.
5
Figure 2 – Effective Length Factor (k) Calculations for Exterior Columns with One Beam Framing into them in
the Direction of Analysis (Sway Frame)
1.9 13.33347.87 22 Consider Slenderness
6.6
uk l
r
ACI 318-14 (6.2.5a)
Where:
1radius of gyration = ( ) (b) 0.3g
g
Ir a or c
A ACI 318-14 (6.2.5.1)
2 2
1 226.35 in.
12 12
g
g
I cr
A
4. Moment Magnification at Ends of Compression Member
A detailed calculation for load combination 4 (gravity plus wind) is shown below to illustrate the procedure. Table
3 summarizes the magnified moment computations for the exterior columns.
2 2 2ns s sM M M ACI 318-14 (6.6.4.6.1b)
Where:
6
1(a)
1
1 moment magnifier (b)
10.75
(c) Second-order elastic analysis
s
u
c
Q
P
P
ACI 318-14 (6.6.4.6.2)
ACI 318-14 (6.6.4.6.2(b)) will be used for comparison purposes with results obtained from spColumn model.
However, (a) and (c) can also be used to calculate the moment magnifier.
∑Pu is the summation of all the factored vertical loads in the first story, and ∑Pc is the summation of the critical
buckling load for all sway-resisting columns in the first story.
2
2
eff
c
u
EIP
kl
ACI 318-14 (6.6.4.4.2)
Where:
0.4(a)
1
0.2(b)
1
(c) 1
c g
ds
c g s se
eff
ds
c
ds
E I
E I E IEI
E I
ACI 318-14 (6.6.4.4.4)
There are three options for calculating the effective flexural stiffness of slender concrete columns (EI)eff. The
second equation provides accurate representation of the reinforcement in the section and will be used in this
example and is also used by the solver in spColumn. Further comparison of the available options is provided in
“Effective Flexural Stiffness for Critical Buckling Load of Concrete Columns” technical note.
4 4422
19,521in.12 12
column
cI ACI 318-14 (Table 6.6.3.1.1(a))
'57,000 57,000 6000 4,415 ksic cE f ACI 318-14 (19.2.2.1.a)
βds is the ratio of maximum factored sustained shear within a story to the maximum factored shear in that story
associated with the same load combination. The maximum factored sustained shear in this example is equal to zero
leading to βds = 0. ACI 318-14 (6.6.3.1.1)
For exterior columns with one beam framing into them in the direction of analysis (12 columns):
With 8-#8 reinforcement equally distributed on all sides and 22 in. x 22 in. column section Ise = 352.6 in.4.
0.2
1
c g s se
eff
ds
E I E IEI
ACI 318-14 (6.6.4.4.4(b))
7
6 20.2 4,415 19,521 29,000 352.627.5 10 kip-in.
1 0effEI
k = 1.9 (calculated previously).
2 6
1 2
27.5 102,933 kip
1.9 13.333cP
For exterior columns with two beams framing into them in the direction of analysis (4 columns):
355 4195.5
70 70
c columns
A
beams
EI
l
EI
l
ACI 318-14 (Figure R6.2.5)
1.0B (Column essentially fixed at base) ACI 318-14 (Figure R6.2.5)
Using Figure R6.2.5 from ACI 318-14 k = 1.71 as shown in the figure below for the exterior columns with
two beams framing into them in the directions of analysis.
Figure 3 – Effective Length Factor (k) Calculations for Exterior Columns with Two Beams Framing into them in the
Direction of Analysis
2 6
2 2
27.5 103,621 kip
1.71 13.333 12cP
For interior columns (8 columns):
8
4 4424
0.7 0.7 19,354 in.12 12
column
cI ACI 318-14 (Table 6.6.3.1.1(a))
'57,000 57,000 6000 4,415 ksic cE f ACI 318-14 (19.2.2.1.a)
For the column below level 2:
34,415 19,354503 10 in.kip
15 20 / 2
c column
c
E I
l
For the column above level 2:
34,415 19,354593 10 in.kip
12
c column
c
E I
l
For beams framing into the columns:
33,605 5,60070 10 in.kip
24
b beam
b
E I
l
Where:
'57,000 57,000 4000 3,605 ksib cE f ACI 318-14 (19.2.2.1.a)
4 4424 20
0.35 0.35 5,600 in.12 12
beam
b hI
ACI 318-14 (Table 6.6.3.1.1(a))
503 5937.8
70 70
c columns
A
beams
EI
l
EI
l
ACI 318-14 (Figure R6.2.5)
1.0B (Column essentially fixed at base) ACI 318-14 (Figure R6.2.5)
Using Figure R6.2.5 from ACI 318-14 k = 1.81 as shown in the figure below for the interior columns.
9
Figure 4 – Effective Length Factor (k) Calculations for Interior Columns
With 8-#8 reinforcement equally distributed on all sides and 24 in. x 24 in. column section Ise = 439.1 in.4.
0.2
1
c g s se
eff
ds
E I E IEI
ACI 318-14 (6.6.4.4.4(b))
6 20.2 4,415 27,648 29,000 439.137.1 10 kip-in.
1 0effEI
2 6
3 2
37.1 104,372 kip
1.81 13.333 12cP
1 1 2 2 3 3c c c cP n P n P n P
12 2,933 4 3,621 8 4,372 84,652 kipcP
For load combination 4:
1.2 1.6 1.2 17,895 1.6 270 21,906 kipu rP D L ASCE 7-10 (2.3.2-3)
1
10.75
s
u
c
P
P
ACI 318-14 (6.6.4.6.2(b))
10
1=1.53
21,9061
0.75 84,652
s
, 1.53 13.7 20.9 ft.kip s Top sM
, ,_ 241.8 20.9 62.7 ft.kip nd Top ns s Top sTop
M M M ACI 318-14 (6.6.4.6.1)
, 1.53 110.4 168.6 ft.kip s Bottom sM
, ,_ 221.1 168.6 189.7 ft.kip nd Bottom ns s Bottom sBottom
M M M ACI 318-14 (6.6.4.6.1)
2 _ 2 _ 2 _ 2 _ 2 2 _1 _1, 189.7 ft.kip 131.5 ft.kipnd nd nd nd st stTop Bottom Bottom Bottom
M max M M M M M
1_ 2 _ 2 _ 2 _ 2 1_1 _1, 62.7 ft.kip 55.4 ft.kipnd nd nd nd st stTop Bottom Top Top
M min M M M M M
Pu = 722.0 kip
A summary of the moment magnification factors and magnified moments for the exterior column for all load
combinations using both equation options ACI 318-14 (6.6.4.4.4(a)) and (6.6.4.4.4(b)) to calculate (EI)eff is
provided in the table below for illustration and comparison purposes. Note: The designation of M1 and M2 is made
based on the second-order (magnified) moments and not based on the first-order (unmagnified) moments.
Table 3 - Factored Axial loads and Magnified Moments for Exterior Column
No. Load Combination Axial Load,
kip
Using ACI 6.6.4.4.4(a) Using ACI 6.6.4.4.4(b)
δs M1, ft-kip M2, ft-kip δs M1, ft-kip M2, ft-kip
1 1.4D 871.4 --- 24.6 48.7 --- 24.6 48.7
2 1.2D + 1.6L + 0.5Lr 869.4 --- 33.4 66.4 --- 33.4 66.4
3 1.2D + 0.5L + 1.6 Lr 797.6 --- 25.0 49.5 --- 25.0 49.5
4 1.2D + 1.6Lr + 0.8W 722.0 1.37 60.6 172.3 1.53 62.7 189.7
5 1.2D + 1.6Lr - 0.8W 799.3 1.37 23.0 -130.1 1.53 20.9 -147.5
6 1.2D + 0.5L + 0.5Lr + 1.6W 710.9 1.39 87.5 330.9 1.55 92.0 367.9
7 1.2D + 0.5L + 0.5Lr - 1.6W 865.4 1.39 11.5 -280.9 1.55 7.0 -317.9
8 0.9D + 1.6W 482.9 1.25 65.5 291.2 1.34 68.0 311.6
9 0.9D - 1.6W 637.4 1.25 -2.9 -259.6 1.34 -5.4 -280.0
11
5. Moment Magnification along Length of Compression Member
In sway frames, second-order effects shall be considered along the length of columns. It shall be permitted to
account for these effects using ACI 318-14 (6.6.4.5) (Nonsway frame procedure), where Cm is calculated using M1
and M2 from ACI 318-14 (6.6.4.6.1) as follows: ACI 318-14 (6.6.4.6.4)
2 2cM M ACI 318-14 (6.6.4.5.1)
Where:
M2 = the second-order factored moment.
magnification factor 1.0
10.75
m
u
c
C
P
P
ACI 318-14 (6.6.4.5.2)
2
2
eff
c
u
EIP
kl
ACI 318-14 (6.6.4.4.2)
Where:
0.4(a)
1
0.2(b)
1
(c) 1
c g
dns
c g s se
eff
dns
c
dns
E I
E I E IEI
E I
ACI 318-14 (6.6.4.4.4)
There are three options for calculating the effective flexural stiffness of slender concrete columns (EI)eff. The second
equation provides accurate representation of the reinforcement in the section and will be used in this example and
is also used by the solver in spColumn. Further comparison of the available options is provided in “Effective
Flexural Stiffness for Critical Buckling Load of Concrete Columns” technical note.
4 4422
19,521in.12 12
column
cI ACI 318-14 (Table 6.6.3.1.1(a))
'57,000 57,000 6000 4,415 ksic cE f ACI 318-14 (19.2.2.1.a)
βdns is the ratio of maximum factored sustained axial load to maximum factored axial load associated with the same
load combination. ACI 318-14 (6.6.4.4.4)
For load combination 4:
, 1.2 622.4 746.9 kipu sustainedP
12
1.2 622.4 1.6 8.6 0.8 48.3 722 kipuP
, 746.91.03 1.00 1.0
722
u sustained
dns dns
u
P
P
355 41911 (Calculated previously)
70
c columns
A
beams
EI
l
EI
l
ACI 318-14 (Figure R6.2.5)
1.0B (Column essentially fixed at base) ACI 318-14 (Figure R6.2.5)
Using Figure R6.2.5(a) from ACI 318-14 k = 0.86 as shown in the figure below for the exterior column.
Figure 5 – Effective Length Factor (k) Calculations for Exterior Column (Nonsway)
With 8-#8 reinforcement equally distributed on all sides and 22 in. x 22 in. column section Ise = 352.6 in.4.
0.2
1
c g s se
eff
dns
E I E IEI
ACI 318-14 (6.6.4.4.4(b))
6 20.2 4,415 19,521 29,000 352.613.7 10 kip-in.
1 1effEI
13
2 6
2
13.7 107,158 kip
0.86 13.333 12cP
For load combination 4:
1.2 622.4 1.6 8.6 0.8 48.3 722 kipuP ASCE 7-10 (2.3.2-3)
1
2
0.6 0.4m
MC
M ACI 318-14 (6.6.4.5.3a)
2 2_ 2189.7 ft.kip (as concluded from section 4)ndM M ACI 318-14 (6.6.4.6.4)
1 1_ 262.7 ft.kip (as concluded from section 4)ndM M ACI 318-14 (6.6.4.6.4)
Since the column is bent in double curvature, M1/M2 is positive. ACI 318-14 (6.6.4.5.3)
62.70.6 0.4 0.468
189.7mC
1.0
10.75
m
u
c
C
P
P
ACI 318-14 (6.6.4.5.2)
0.468=0.54 1.00 1.00
7221
0.75 7,158
min 0.6 0.03uM P h ACI 318-14 (6.6.4.5.4)
Where Pu = 722 kip, and h = the section dimension in the direction being considered = 22 in.
min
0.6 0.03 22722 75.81 ft.kip
12M
1 min 162.70 ft.kip 75.81ft.kip 75.81 ft.kipM M M ACI 318-14 (6.6.4.5.4)
1 1cM M ACI 318-14 (6.6.4.5.1)
1 1.00 75.81 75.81 ft.kipcM
2 2,min 2189.7 ft.kip 75.81ft.kip 189.7 ft.kipM M M ACI 318-14 (6.6.4.5.4)
2 2cM M ACI 318-14 (6.6.4.5.1)
2 1.00 189.7 189.7 ft.kipcM
Mc1 and Mc2 will be considered separately to ensure proper comparison of resulting magnified moments against
negative and positive moment capacities of unsymmetrical sections as can be seen in the following figure.
14
Figure 6 – Column Interaction Diagram for Unsymmetrical Section
A summary of the moment magnification factors and magnified moments for the exterior column for all load
combinations using both equation options ACI 318-14 (6.6.4.4.4(a)) and (6.6.4.4.4(b)) to calculate (EI)eff is
provided in the table below for illustration and comparison purposes.
Table 4 - Factored Axial loads and Magnified Moments along Exterior Column Length
No. Load Combination Axial Load,
kip
Using ACI 6.6.4.4.4(a) Using ACI 6.6.4.4.4(b)
δ Mc1,
ft-kip
Mc2,
ft-kip δ
Mc1,
ft-kip
Mc2,
ft-kip
1 1.4D 871.4 1.00 91.5 91.5 1.00 91.5 91.5
2 1.2D + 1.6L + 0.5Lr 869.4 1.00 91.3 91.3 1.00 91.3 91.3
3 1.2D + 0.5L + 1.6 Lr 797.6 1.00 83.7 83.7 1.00 83.7 83.7
4 1.2D + 1.6Lr + 0.8W 722.0 1.00 75.8 172.3 1.00 75.8 189.7
5 1.2D + 1.6Lr - 0.8W 799.3 1.00 83.9 -130.1 1.00 83.9 -147.5
6 1.2D + 0.5L + 0.5Lr + 1.6W 710.9 1.00 87.5 330.9 1.00 92.0 367.9
7 1.2D + 0.5L + 0.5Lr - 1.6W 865.4 1.00 90.9 -280.9 1.00 90.9 -317.9
8 0.9D + 1.6W 482.9 1.00 65.5 291.2 1.00 68.0 311.6
9 0.9D - 1.6W 637.4 1.00 66.9 -259.6 1.00 66.9 -280.0
For column design ACI 318 requires the second-order moment to first-order moment ratios should not exceed 1.40.
If this value is exceeded, the column design needs to be revised. ACI 318-14 (6.2.6)
15
Table 5 - Second-Order Moment to First-Order Moment Ratios
No. Load Combination Using ACI 6.6.4.4.4(a) Using ACI 6.6.4.4.4(b)
Mc1/M1(1st) Mc2/M2(1st) Mc1/M1(1st) Mc2/M2(1st)
1 1.4D 1.00* 1.00* 1.00* 1.00*
2 1.2D + 1.6L + 0.5Lr 1.00* 1.00* 1.00* 1.00*
3 1.2D + 0.5L + 1.6 Lr 1.00* 1.00* 1.00* 1.00*
4 1.2D + 1.6Lr + 0.8W 1.00* 1.31 1.00* 1.40 < 1.44
5 1.2D + 1.6Lr - 0.8W 1.00* 1.40 < 1.46 1.00* 1.40 < 1.65
6 1.2D + 0.5L + 0.5Lr + 1.6W 1.14 1.35 1.20 1.40 < 1.50
7 1.2D + 0.5L + 0.5Lr - 1.6W 1.00* 1.40 < 1.43 1.00* 1.40 < 1.62
8 0.9D + 1.6W 1.12 1.23 1.16 1.32
9 0.9D - 1.6W 1.00* 1.27 1.00* 1.37 * Cutoff value of Mmin is applied to M1(1st) and M2(1st) in order to avoid unduly large ratios in cases where M1(1st) and M2(1st)
moments are smaller than Mmin.
16
6. Column Design
Based on the factored axial loads and magnified moments considering slenderness effects, the capacity of the
assumed column section (22 in. x 22 in. with 8-#8 bars distributed all sides equal) will be checked and confirmed
to finalize the design. A column interaction diagram will be generated using strain compatibility analysis, the
detailed procedure to develop column interaction diagram can be found in “Interaction Diagram – Tied Reinforced
Concrete Column” example.
The axial compression capacity ϕPn for all load combinations will be set equals to Pu, then the moment capacity
ϕMn associated to ϕPn will be compared with the magnified applied moment Mu. The design check for load
combination #4 is shown below for illustration. The rest of the checks for the other load combinations are shown
in the following Table.
Figure 7 – Strains, Forces, and Moment Arms (Load Combination 4)
The following procedure is used to determine the nominal moment capacity by setting the design axial load
capacity, ϕPn, equal to the applied axial load, Pu and iterating on the location of the neutral axis.
6.1. c, a, and strains in the reinforcement
Try 12.75 in.c
Where c is the distance from the fiber of maximum compressive strain to the neutral axis.
ACI 318-14 (22.2.2.4.2)
1 0.75 12.75 9.563 in.a c ACI 318-14 (22.2.2.4.1)
Where:
'
1
0.05 4000 0.05 6000 40000.85 0.85 0.75
1000 1000
cf
ACI 318-14 (Table 22.2.2.4.3)
0.003cu ACI 318-14 (22.2.2.1)
17
600.00207
29,000
y
y
s
f
E
1
0.003 0.003( ) (19.625 12.75) 0.00162 (Tension) <
12.75s yd c
c
tension reinforcement has not yielded
0.65 ACI 318-14 (Table 21.2.2)
'
1 2
0.003 0.003( ) (12.75 2.375) 0.00244 (Compression) >
12.75s yc d
c
'
2
0.003 0.003(12.75 11) 0.00041(Compression) <
2 12.75s y
hc
c
6.2. Forces in the concrete and steel
'0.85 0.85 6,000 9.563 22 1073 kipc cC f a b ACI 318-14 (22.2.2.4.1)
0.00162 29,000,000 46,912 psis s sf E
1T 46,912 3 0.79 111.2 kips y sf A
'
1Since > compression reinforcement has yieldeds y
'
1 60,000 psis yf f
'
2Since < compression reinforcement has not yieldeds y
' '
2 2 0.00041 29,000,000 11,941 psis s sf E
The area of the reinforcement in this layer has been included in the area (ab) used to compute Cc. As a result,
it is necessary to subtract 0.85fc’ from fs’ before computing Cs:
' ' '
1 1 1C 0.85 60,000 0.85 6,000 3 0.79 130.1 kips s c sf f A
' ' '
2 2 2C 0.85 11,941 0.85 6,000 2 0.79 18.9 kips s c sf f A
6.3. ϕPn and ϕMn
1 2 1,073 130.1 18.9 111.2 1,111kipn c s s sP C C C T
0.65 1,111 722 kip = n uP P
The assumption that c = 12.75 in. is correct
18
1 2 2 12 2 2 2 2 2
n c s s s
h a h h h hM C C d C T d
22 9.563 22 22 22 221,073 130.1 2.375 18.9 111.2 19.625 729kip.ft
2 2 2 2 2 2nM
20.65 729 474kip.ft 189.7 kip.ftn u cM M M
Table 6 – Exterior Column Axial and Moment Capacities
No. Pu, kip Mu = M2(2nd), ft-kip c, in. εt = εs φ φPn, kip φMn, kip.ft
1 871.4 91.5 14.85 0.00096 0.65 871.4 459.4
2 869.4 91.3 14.85 0.00097 0.65 869.4 459.7
3 797.6 83.7 13.75 0.00128 0.65 797.6 468.2
4 722.0 189.7 12.75 0.00162 0.65 722.0 474.1
5 799.3 -147.5 13.78 0.00127 0.65 799.3 468.0
6 710.9 367.9 12.61 0.00167 0.65 710.9 474.8
7 865.4 -317.9 14.76 0.00099 0.65 865.4 460.2
8 482.9 311.6 7.36 0.005 0.9 482.9 557.2
9 637.4 -280.0 11.68 0.00204 0.65 637.4 478.8
Therefore, since ϕMn > Mu for all ϕPn = Pu, use 22 x 22 in. column with 8-#8 bars.
19
7. Column Interaction Diagram - spColumn Software
spColumn program performs the analysis of the reinforced concrete section conforming to the provisions of the
Strength Design Method and Unified Design Provisions with all conditions of strength satisfying the applicable
conditions of equilibrium and strain compatibility and includes slenderness effects using moment magnification
method for sway and nonsway frames. For this column section, we ran in investigation mode with control points
using the 318-14. In lieu of using program shortcuts, spSection (Figure 8) was used to place the reinforcement
and define the cover to illustrate handling of irregular shapes and unusual bar arrangement.
Figure 8 – spColumn Model Editor (spSection)
21
Figure 10 – Column Section Interaction Diagram about the X-Axis – Design Check for Load Combination 4
(spColumn)
22
23
24
25
26
27
28
8. Summary and Comparison of Design Results
Table 7 - Factored Axial loads and Magnified Moments at Column Ends Comparison
No. Pu, kip δs M1(2nd), ft-kip M2(2nd), ft-kip
Hand Reference spColumn Hand Reference spColumn Hand Reference spColumn Hand Reference* spColumn
1 871.4 871.4 871.4 N/A N/A N/A 24.6 24.6 24.6 48.7 48.7 48.7
2 869.4 869.4 869.4 N/A N/A N/A 33.4 33.4 33.4 66.4 66.4 66.4
3 797.6 797.6 797.6 N/A N/A N/A 25.0 25.0 25.0 49.5 49.5 49.5
4 722.0 722.0 722.0 1.53 1.38 1.53 62.7 60.6 62.7 189.7 173.5 189.7
5 799.3 799.3 799.3 1.53 1.38 1.53 20.9 23.0 20.9 -147.5 -131.3 -147.4
6 710.9 710.9 7110.9 1.55 1.39 1.55 92.0 87.5 92.0 367.9 331.9 367.8
7 865.4 865.4 865.4 1.55 1.39 1.55 7.0 11.5 7.0 -317.9 -281.9 -317.9
8 482.9 482.9 482.9 1.34 1.25 1.34 68.0 65.5 68.0 311.6 292.0 311.7
9 637.4 637.4 637.4 1.34 1.25 1.34 -5.4 -2.9 -5.3 -280.0 -260.3 280.0
Table 8 - Magnified Moments along Column Length to First-Order Moment Ratios Comparison
No. δ Mc1, ft-kip Mc2, ft-kip Mc1/M1(1st) Mc2/M2(1st)
Hand Reference* spColumn Hand Reference* spColumn Hand Reference* spColumn Hand Reference* spColumn Hand Reference* spColumn
1 1.00 --- 1.00 91.5 --- 91.5 91.5 --- 91.5 1.00 --- 1.00 1.00 --- 1.00
2 1.00 --- 1.00 91.3 --- 91.3 91.3 --- 91.3 1.00 --- 1.00 1.00 --- 1.00
3 1.00 --- 1.00 83.7 --- 83.7 83.7 --- 83.7 1.00 --- 1.00 1.00 --- 1.00
4 1.00 --- 1.00 75.8 --- 75.8 189.7 --- 189.7 1.00 --- 1.00 1.44 --- 1.44
5 1.00 --- 1.00 83.9 --- 83.9 -147.5 --- -147.4 1.00 --- 1.00 1.65 --- 1.65
6 1.00 --- 1.00 92.0 --- 92.0 367.9 --- 367.8 1.20 --- 1.20 1.50 --- 1.50
7 1.00 --- 1.00 90.9 --- 90.9 -317.9 --- -317.9 1.00 --- 1.00 1.62 --- 1.62
8 1.00 --- 1.00 68.0 --- 68.0 311.6 --- 311.7 1.16 --- 1.16 1.32 --- 1.32
9 1.00 --- 1.00 -66.9 --- -66.9 -280.0 --- 280.0 1.00 --- 1.00 1.37 --- 1.37 * Moment magnification along the length of the column is not covered by the reference
29
Table 9 - Design Parameters Comparison
No. c, in. εt = εs φ φPn, kip φMn, kip.ft
Hand Reference* spColumn Hand Reference* spColumn Hand Reference* spColumn Hand Reference* spColumn Hand Reference* spColumn
1 14.85 14.85 14.85 0.00096 0.00096 0.00096 0.65 0.65 0.65 871.4 871.4 871.4 459.4 459.4 459.4
2 14.85 14.82 14.85 0.00097 0.00097 0.00097 0.65 0.65 0.65 869.4 869.4 869.4 459.7 459.7 459.7
3 13.75 13.75 13.75 0.00128 0.00128 0.00128 0.65 0.65 0.65 797.6 797.6 797.6 468.2 468.2 468.2
4 12.75 12.75 12.75 0.00162 0.00162 0.00162 0.65 0.65 0.65 722.0 722.0 722.0 474.1 474.1 474.1
5 13.78 13.78 13.78 0.00127 0.00127 0.00127 0.65 0.65 0.65 799.3 799.3 799.3 468.0 468.0 468.0
6 12.61 12.61 12.61 0.00167 0.00167 0.00167 0.65 0.65 0.65 710.9 710.9 7110.9 474.8 474.8 474.8
7 14.76 14.76 14.76 0.00099 0.00099 0.00099 0.65 0.65 0.65 865.4 865.4 865.4 460.2 460.2 460.2
8 7.36 7.36 7.36 0.00500 0.00500 0.00500 0.90 0.90 0.90 482.9 482.9 482.9 557.2 557.2 557.2
9 11.68 11.68 11.68 0.00204 0.00204 0.00204 0.65 0.65 0.65 637.4 637.4 637.4 478.8 478.8 478.8 * Notes on ACI 318-11 Building Code Requirements for Structural Concrete, Twelfth Edition, 2013 Portland Cement Association, Example 11-2
In all of the hand calculations and the reference used illustrated above, the results are in precise agreement with the automated exact results obtained from the
spColumn program.
30
9. Conclusions & Observations
The analysis of the reinforced concrete section performed by spColumn conforms to the provisions of the Strength
Design Method and Unified Design Provisions with all conditions of strength satisfying the applicable conditions
of equilibrium and strain compatibility and includes slenderness effects using moment magnification method for
sway and nonsway frames.
ACI 318 provides multiple options for calculating values of k, (EI)eff , δs, and δ leading to variability in the
determination of the adequacy of a column section. Engineers must exercise judgment in selecting suitable options
to match their design condition as is the case in the reference where the author conservatively made assumptions
to simplify and speed the calculation effort. The spColumn program utilizes the exact methods whenever possible
and allows user to override the calculated values with direct input based on their engineering judgment wherever
it is permissible.
In load combinations 4 to 7, Mu including second-order effects exceeds 1.4 Mu due to first-order effects (see Table
5). This indicates that in this building, the weight of the structure is high in proportion to its lateral stiffness
leading to excessive PΔ effect (secondary moments are more than 25 percent of the primary moments). The PΔ
effects will eventually introduce singularities into the solution to the equations of equilibrium, indicating physical
structural instability. It was concluded in the literature that the probability of stability failure increases rapidly
when the stability index Q exceeds 0.2, which is equivalent to a secondary-to-primary moment ratio of 1.25. The
maximum value of the stability coefficient θ (according to ASCE/SEI 7) which is close to stability coefficient Q
(according to ACI 318) is 0.25. The value 0.25 is equivalent to a secondary-to-primary moment ratio of 1.33.
Hence, the upper limit of 1.4 on the secondary-to-primary moment ratio was selected by the ACI 318.
The moment magnification factor values δs calculated in this document and spColumn are different from the
values calculated by the reference. ACI 318 provides three equation options to calculate the effective stiffness
modulus (EI)eff as was discussed previously in this document. Equation 6.6.4.4.4(b) is more accurate than equation
6.6.4.4(a) but is more difficult to use because Ise is not known until reinforcement is chosen. The reference used
equation 6.6.4.4(a) due to its simplicity while spColumn uses equation 6.6.4.4.4(b) since an iterative procedure
is used to select the optimum reinforcement configuration.
As can be seen in Table 5 of this example, exploring the impact of other code permissible equation options
provides the engineer added flexibility in decision making regarding design. For load combinations 4 - 7 resolving
the stability concern may be viable through a frame analysis providing values for Vus and Δo to calculate
magnification factor δs and may allow the proposed design to be acceptable. Creating a complete model with
detailed lateral loads and load combinations to account for second order effects may not be warranted for all cases
of slender column design nor is it disadvantageous to have a higher margin of safety when it comes to column
slenderness and frame stability considerations.